Theorem spw 2036

Description: Weak version of the specialization scheme sp 2175. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2175 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2175 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2130 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2175 are spfw 2035 (minimal distinct variable requirements), spnfw 1982 (when 𝑥 is not free in ¬ 𝜑), spvw 1983 (when 𝑥 does not appear in 𝜑), sptruw 1807 (when 𝜑 is true), spfalw 2000 (when 𝜑 is false), and spvv 1999 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1912	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1912	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1912	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2035	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781

This theorem is referenced by:  hba1w  2049  19.8aw  2052  exexw  2053  spaev  2054  ax12w  2128  nfcriOLD  2892  rspw  3232  reldisj  4451  ralidmw  4507  dtruALT2  5368  dtruOLD  5441  bj-ssblem1  35847  bj-ax12w  35870